New lower bounds on the size-Ramsey number of a path

Abstract

We prove that for all graphs with at most $(3.75-o(1))n$ edges there exists a 2-coloring of the edges such that every monochromatic path has order less than $n$. This was previously known to be true for graphs with at most $2.5n-7.5$ edges. We also improve on the best-known lower bounds in the $r$-color case.

arXiv comment18 pages, 5 figures; (v4) Updated in response to referee comments. Simplified the proof of Lemma 2.4. Improved the exposition of the proofs of Lemma 2.16 and Theorem 1.1. (v3) Significantly reorganized the proof of the main theorem (v2) Minor edits